Im not sure if anyone could think this build is trash at first sight, so anyways im gonna explain why its waaay better than BoRK lucian although i may say that im no expert and not all my points are 100% correct, anyways just test this build its some go0d ****.
1. BoRK build is DEFENSIVE
Although BoRK is an offensive item, its obviously not as offensive as an infinity edge, so bork's damage is good early, but will fall off withouth more damage items, so when you buy a black cleaver with the standard build, its not enough damage; you're building too much defensiveness, sure cdr is nice but if dont have enough damage to use it at its full potential its useless.
2. BoRK gives USELESS stats(For lucian)
Lucian abuses the fk out of early AD, thing that BoRK doesnt give, at least not as much as BF Sword. Instead it gives you attack speed, which is a good stat, but one that doesnt change much in lucians early game which is the stage where you should start getting a lead, it gives you 12% lifesteal which is nice, but as i said you dont need defensiveness, as an ADC your role is to build full OFFENSIVE, defensiveness should come later into the game when you have more damage and you can abuse lifesteal better.
3. CRIT damage
Imagine you typical lucian with 2 items, his black cleaver and his bork as usual. Now his damage its usually at the 200-250 and that if he manages to break armor. Now, imagine infinity edge lucian that has more ad, is almost equally as tanky and can hit crits for 500 at the 2 item mark, can other adc do the same thing? (infinity and black cleaver give you 200 ad at around that time in the game if not that more)
4. This build is not GREEDY
Not sure why someone would think that this build is greedy as you still have a black cleaver thing that most adcs dont use, but with the new lifesteal rune if fed you become a train of destruction that has mobility, damage, survivavility with your cleaver and 11% lifesteal that you gain off your rune. (8% + 3% from dorans)
5. BoRK is a good ITEM
Its really good for shredding tanks, but not all champions are tanks and although your passive abuses it, its still an item that should be built later into the game. You can switch you bloodthirster for it later if you need more damage.

As any build, it has its clear ups and downs, which ill explain.
1. Why IE?
Damage. It is indeed obvious that there is a massive damage difference betwen these two, especially with a close range adc like lucian you NEED enough damage to where when you go in you know you can burst the other adc. Some may say that the damage difference is not that massive, but when you factor in the ad that cleaver( compared to a crit item which essence reaver builds) gives you, and the 2.5 multiplier I´d say its more than noticeable, at the 15 min mark you'll be criting for 500 damage. If you had essence reaver you'd be criting for around 320-340 withouth the armor penetration that amplifies damage of your spells.
2. Why sticking to cleaver?
As I previously mentioned, with lucian you need to know your damage, but also you have to know how much damage you can endure, cleaver gives you exactly what you need, it amplifies damage from you spells, ad that you abuse early on, movement speed, but most importantly; it gives you the survivavility that you need to start pumping damage, because cdr wont save you from a rengar or a talon or something like that, just that 400 extra health will allow you to deal damage knowing that you wont be just bursted down with an ability or 2. Obviously cleaver doesnt make a tank, it just gives you 400 extra health that other adcs dont have, with the armor pen and bigass crits its an excelent dueling item.
3. What?
If I havent made my point clear, ill do it now. Reaver's damage falls off at the 2 item mark making you both really squishy and with weak damage for someone as close ranged as lucian is. With games ending at the usually at the 20-30 min mark maintining good presence throughout is really important which is the most important point of this build.
4. The good things
This build is one that doesnt have extreme powerspikes like reaver has, instead it gives you SOLIDNESS throughout all the game. If I were to say a really noticeable powerspike with this build is at the 3 item mark (thing that most adcs do but its really noticeable with this build) because you have both your 500 crits, attack speed, crit chance, cdr, health and movement speed. I'd say if you have experience with lucia, in my opinion this would be a better build as it allows you to have more presence in the map than reaver lucian. So, to sum it up if you wanna carry yourself or you team/support/peeler sucks *** this build is excelent, it allows you self peel the best of any. What you excel at is destroying squishies, tanks will give you a lot of problem but your absolute focus should be to look an oportunity to burst the enemy adc or main damage source. This build should give you enough survivavility to deal just enough damage.
5. The bad things
As I said, this build is SOLID all the game which means you cant abuse powerspikes that much, and you will kinda fall off late game, at least you wont do as much damage as reaver lucian (if you know how to play lucian you should play to your advantages) so, it is more difficult to deal damage if the game goes extremely late, tanks will be your main problem, but this build is extremely flexible and you can change your rapidfire for a last whisper.

If you like spellcasting nature, slow but hard hitting basic attacks, balanced stats and you know how to make you combos, this is you build. Its focused on burst damage, not continous damage as reaver is. Reaver has a lot more dps late game but in my opinion thats not what lucian is designed for, anyways I encourage you to try this build, im sure you wont be disapointed. Basically, if you like to just autoattack and destroy everything, reaver is more your style, if you like casting spells and moving around this is more your style.